MATH 10560: CALCULUS II
TRIGONOMETRIC FORMULAS
Basic Identities
The functions cos(θ) and sin(θ) are defined to be the x and y coordinates of the point at an angle of θ
on the unit circle. Therefore, sin(−θ) = − sin(θ), cos(−θ) = cos(θ), and sin2 (θ) + cos2 (θ) = 1. The other
trigonometric functions are defined in terms of sine and cosine:
tan(θ) = sin(θ)/ cos(θ)
sec(θ) = 1/ cos(θ)
cot(θ) =
csc(θ) =
cos(θ)/ sin(θ) = 1/ tan(θ)
1/ sin(θ)
Dividing sin2 (θ) + cos2 (θ) = 1 by cos2 (θ) or sin2 (θ) gives tan2 (θ) + 1 = sec2 (θ) and 1 + cot2 (θ) = csc2 (θ).
Addition Formulas
The following two addition formulas are fundamental:
sin(A + B)
cos(A + B)
= sin(A) cos(B) + cos(A) sin(B)
= cos(A) cos(B) − sin(A) sin(B)
They can be used to prove simple identities like sin(π/2 − θ) = sin(π/2) cos(θ) + cos(π/2) sin(θ) = cos(θ), or
cos(x − π) = cos(x) cos(π) − sin(x) sin(π) = − cos(x). If we set A = B in the addition formulas we get the
double-angle formulas:
sin(2A) = 2 sin(A) cos(A)
cos(2A) = cos2 (A) − sin2 (A)
The formula for cos(2A) is often rewritten by replacing cos2 (A) with 1 − sin2 (A) or replacing sin2 (A) with
1 − cos2 (A) to get
cos(2A) = 1 − 2 sin2 (A)
cos(2A) = 2 cos2 (A) − 1
Solving for sin2 (A) and cos2 (A) yields identities important for integration:
1
1
sin2 (A) = (1 − cos(2A))
cos2 (A) = (1 + cos(2A))
2
2
The addition formulas can also be combined to give other formulas important for integration:
sin A sin B
=
cos A cos B
=
sin A cos B
=
1
2 [cos(A − B) − cos(A + B)]
1
2 [cos(A − B) + cos(A + B)]
1
2 [sin(A − B) + sin(A + B)]
Derivatives and Integrals
sin′ (x) = cos(x)
cos′ (x) = − sin(x)
tan′ (x) = sec2 (x)
R
R sin(x) dx = − cos(x) + C
R cos(x) dx = sin(x) + C
tan(x) dx = ln | sec(x)| + C
sec′ (x) = sec(x) tan(x)
csc′ (x) = − csc(x) cot(x)
cot′ (x) = − csc2 (x)
R
R sec(x) dx = ln | sec(x) + tan(x)| + C
R csc(x) dx = ln | csc(x) − cot(x)| + C
cot(x) dx = − ln | csc(x)| + C